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Since real materials may not realize the ideal Heisenberg model, more
general spin Hamiltonians have been investigated, mainly with
numerical simulations [65,66,67,68].
From these works, a phase diagram has been obtained for the
Hamiltonian with the XXZ-type interaction and a uniaxial single ion
anisotropy D:
The phase diagram is shown in Fig.36 [68].
It has been found that the Haldane phase (H) exist in a relatively large
parameter region, which includes the pure Heisenberg model (
).
Figure:
Phase diagram of the Hamiltonian eq.42 is shown
(cite from [68]). The symbols are: (H) Haldane phase,
(N) Néel ordered phase, (XY) XY phase, (F) Ferromagnetic phase
and (D) large-D phase.
 |
The effect of the inter-chain interactions has also been
investigated [69,70,71].
Fig.37 shows the phase diagram [71] obtained from
a numerical simulation of the Hamiltonian:
where (i,j) denotes the nearest inter-chain neighboring pair.
It has been found that the Haldane phase (H) survives in the
presence of the inter-chain interaction (J'), if the single ion anisotropy (D)
is small.
Figure:
Phase diagram of the Hamiltonian (eq.43) is shown
(cite from [71]). The symbols are: (H) Haldane phase,
(N) Néel ordered phase, (XY) XY phase, and (D) large-D phase.
 |
The above introduced theoretical works all supported Haldane's
conjecture. The next section introduces previous experimental results
of several S=1 spin systems, which were studied to test Haldane's conjecture in real
materials.
Next: 5.1.4 Experimental evidence for
Up: 5.1 Introduction
Previous: 5.1.2 The Valence-Bond-Solid Hamiltonian